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 Page 1


Laplace Transform And Its 
Applications
Page 2


Laplace Transform And Its 
Applications
Topics
? Definition of Laplace Transform
? Linearity of the Laplace Transform
? Laplace Transform of some Elementary Functions
? First Shifting Theorem
? Inverse Laplace Transform
? Laplace Transform of  Derivatives & Integral
? Differentiation & Integration of Laplace Transform
? Evaluation of Integrals By Laplace Transform
? Convolution Theorem
? Application to Differential Equations
? Laplace Transform of Periodic Functions
? Unit Step Function
? Second Shifting Theorem
? Dirac Delta Function
Page 3


Laplace Transform And Its 
Applications
Topics
? Definition of Laplace Transform
? Linearity of the Laplace Transform
? Laplace Transform of some Elementary Functions
? First Shifting Theorem
? Inverse Laplace Transform
? Laplace Transform of  Derivatives & Integral
? Differentiation & Integration of Laplace Transform
? Evaluation of Integrals By Laplace Transform
? Convolution Theorem
? Application to Differential Equations
? Laplace Transform of Periodic Functions
? Unit Step Function
? Second Shifting Theorem
? Dirac Delta Function
Definition of Laplace Transform
? Let f(t) be a given function of t defined for all 
then the Laplace Transform ot f(t) denoted by L{f(t)}
or         or F(s) or          is defined as
provided the integral exists,where s is a parameter real 
or complex.
0 ? t
) (s f ) (s ?
dt t f e s s F s f t f L
st
) ( ) ( ) ( ) ( )} ( {
0
?
?
?
? ? ? ? ?
Page 4


Laplace Transform And Its 
Applications
Topics
? Definition of Laplace Transform
? Linearity of the Laplace Transform
? Laplace Transform of some Elementary Functions
? First Shifting Theorem
? Inverse Laplace Transform
? Laplace Transform of  Derivatives & Integral
? Differentiation & Integration of Laplace Transform
? Evaluation of Integrals By Laplace Transform
? Convolution Theorem
? Application to Differential Equations
? Laplace Transform of Periodic Functions
? Unit Step Function
? Second Shifting Theorem
? Dirac Delta Function
Definition of Laplace Transform
? Let f(t) be a given function of t defined for all 
then the Laplace Transform ot f(t) denoted by L{f(t)}
or         or F(s) or          is defined as
provided the integral exists,where s is a parameter real 
or complex.
0 ? t
) (s f ) (s ?
dt t f e s s F s f t f L
st
) ( ) ( ) ( ) ( )} ( {
0
?
?
?
? ? ? ? ?
Linearity of the Laplace Transform
? If  L{f(t)}=        and                            then for any 
constants a and b 
) (s f
) ( )] ( [ s g t g L ?
)] ( [ )] ( [ )] ( ) ( [ t g bL t f aL t bg t af L ? ? ?
)] ( [ )] ( [ )} ( ) ( {
) ( ) (                           
)] ( ) ( [ )} ( ) ( {
Definition -By : Proof
0 0
0
t g bL t f aL t bg t af L
dt t g e b dt t f e a
dt t bg t af e t bg t af L
st st
st
? ? ?
? ?
? ? ?
? ?
?
?
?
?
?
?
?
Page 5


Laplace Transform And Its 
Applications
Topics
? Definition of Laplace Transform
? Linearity of the Laplace Transform
? Laplace Transform of some Elementary Functions
? First Shifting Theorem
? Inverse Laplace Transform
? Laplace Transform of  Derivatives & Integral
? Differentiation & Integration of Laplace Transform
? Evaluation of Integrals By Laplace Transform
? Convolution Theorem
? Application to Differential Equations
? Laplace Transform of Periodic Functions
? Unit Step Function
? Second Shifting Theorem
? Dirac Delta Function
Definition of Laplace Transform
? Let f(t) be a given function of t defined for all 
then the Laplace Transform ot f(t) denoted by L{f(t)}
or         or F(s) or          is defined as
provided the integral exists,where s is a parameter real 
or complex.
0 ? t
) (s f ) (s ?
dt t f e s s F s f t f L
st
) ( ) ( ) ( ) ( )} ( {
0
?
?
?
? ? ? ? ?
Linearity of the Laplace Transform
? If  L{f(t)}=        and                            then for any 
constants a and b 
) (s f
) ( )] ( [ s g t g L ?
)] ( [ )] ( [ )] ( ) ( [ t g bL t f aL t bg t af L ? ? ?
)] ( [ )] ( [ )} ( ) ( {
) ( ) (                           
)] ( ) ( [ )} ( ) ( {
Definition -By : Proof
0 0
0
t g bL t f aL t bg t af L
dt t g e b dt t f e a
dt t bg t af e t bg t af L
st st
st
? ? ?
? ?
? ? ?
? ?
?
?
?
?
?
?
?
Laplace Transform of some Elementary 
Functions
a s if 
a - s
1
) (
                                    
e . ) e (      
Definition -By : Proof      
a - s
1
) L(e  (2)
) 0 ( ,
s
1
1 . ) 1 (      
Definition -By : Proof      
s
1
L(1)  (1)
0
) (
0
) (
0
at at
at
0
0
? ?
?
?
?
?
?
?
? ?
?
? ?
?
? ?
?
?
?
?
?
?
?
? ?
?
?
? ?
?
? ?
?
?
?
?
?
?
? ?
?
a s
e
dt e dt e L
s
s
e
dt e L
t a s
t a s st
st
st
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FAQs on PPT: Laplace Transform & Its Applications - Signals and Systems - Electrical Engineering (EE)

1. What is the Laplace transform and how is it used?
The Laplace transform is a mathematical tool used to transform a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations and analyze dynamic systems. By applying the Laplace transform, we can convert differential equations into algebraic equations, making it easier to solve and analyze the behavior of systems.
2. What are the advantages of using the Laplace transform?
The Laplace transform has several advantages in solving differential equations and analyzing systems. Firstly, it simplifies the solution process by converting differential equations into algebraic equations, which are generally easier to solve. Secondly, it provides a powerful method for analyzing the behavior of systems in the frequency domain, allowing us to understand system responses to different inputs more intuitively. Additionally, the Laplace transform can handle a wide range of functions, including functions with discontinuities, making it a versatile tool in various fields of science and engineering.
3. Can the Laplace transform be used for both linear and nonlinear systems?
Yes, the Laplace transform can be applied to both linear and nonlinear systems. However, it is more commonly used for linear systems, where the principle of superposition holds. In the case of nonlinear systems, the Laplace transform can still be useful in specific situations, such as linearizing the system around an operating point or analyzing small signal behavior.
4. What are some common applications of the Laplace transform?
The Laplace transform has a wide range of applications in various fields. Some common applications include: - Electrical circuit analysis: The Laplace transform is used to analyze the behavior of electrical circuits, such as calculating voltage and current responses to different inputs. - Control systems: The Laplace transform is essential in analyzing and designing control systems, helping engineers understand system stability, transient response, and frequency response. - Signal processing: The Laplace transform is used in signal processing to analyze and manipulate signals in the frequency domain, such as filtering, noise reduction, and modulation. - Vibrations and mechanical systems: The Laplace transform is used to analyze the dynamic behavior of vibrating systems and mechanical systems, including calculating natural frequencies, transient response, and steady-state response. - Fluid dynamics: The Laplace transform is also applied in fluid dynamics to solve partial differential equations and analyze the behavior of fluid flows.
5. Are there any limitations or drawbacks of using the Laplace transform?
While the Laplace transform is a powerful tool, it does have some limitations. One limitation is that it is primarily applicable to linear time-invariant systems, where the system parameters do not change with time. Nonlinear systems can be more challenging to analyze using the Laplace transform, and additional techniques may be required. Another limitation is that the Laplace transform assumes the existence of the transform for a given function, which may not always be possible for certain functions. Additionally, the Laplace transform may introduce complexity in the form of complex variables and complex frequency domain representations, which can be more difficult to interpret compared to the time domain.
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